There are no Diophantine quadruples of Pell numbers

In this paper, we prove that there are no positive integers [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] such that [Formula: see text] is a Diophantine quadruple, where for a positive integer [Formula: see text], [Formula: see text] is the [Formula: see text]th Pell number.

On Generalized Pell Numbers of Order r ≥ 2

In this paper we investigate the generalized Pell numbers of order r ≥ 2 through the properties of their related fundamental system of generalized Pell numbers. That is, the generalized Pell number of order r ≥ 2; are expressed as a linear combination of a fundamental system of generalized Pell numbers. The properties of this fundamental system are examined and results can be established for generalized Pell numbers of order r ≥ 2. Some identities and combinatorial results are established. Moreover, the analytic study of the fundamental system of generalized Pell is provided. Furthermore, the generalized Pell Cassini identity type is provided.

On Pillai's problem with Pell numbers and powers of 2

International audience In this paper, we find all integers c having at least two representations as a difference between a Pell number and a power of 2.

Gaussian Pell Numbers

Gaussian numbers means representation as Complex numbers. In this work, Gaussian Pell numbers are defined from recurrence relation of Pell numbers. Here the recurrence relation on Gaussian Pell number is represented in two dimensional approach. This provides an extension of Pell numbers into the complex plane.

Enumeration of Standard Young Tableaux of Shifted Strips with Constant Width

Let $g_{n_1,n_2}$ be the number of standard Young tableau of truncated shifted shape with $n_1$ rows and $n_2$ boxes in each row. By using the integral method this paper derives the recurrence relations of $g_{3,n}$, $g_{n,4}$ and $g_{n,5}$ respectively. Specifically, $g_{n,4}$ is the $(2n-1)$-st Pell number.

Pell numbers whose Euler function is a Pell number

We show that the only Pell numbers whose Euler function is also a Pell number are 1 and 2.